Question: Find the sum of the geometric series $1 + 1.2 + 1.2^2 +1.2^3 +... + 1.2^{74}$ Choose 1 answer: Choose 1 answer: (Choice A) A $394{,}612.9$ (Choice B) B $723{,}456.14$ (Choice C) C $3{,}617{,}275.71$ (Choice D) D $ 4{,}340{,}731.85 $
Getting started We're dealing with a geometric series because each term is multiplied by $1.2$ to get the next term. We need a formula to compute the sum of the terms. Formula for geometric series The sum $S_n$ of a finite geometric series is $S_n = \dfrac{a_1(1-r^n)}{1-r}$ where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = {1})$ is given in the question. The number of terms $n$ is ${75}$ because there are ${75}$ numbers from $0$ to $74$. [Where do the 0 and 74 come from?] The common ratio $r$ is ${1.2}$ because each term is multiplied by ${1.2}$ to get the next term. [How did we find the common ratio r?] Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac{a_1(1-r^n)}{1-r} \\\\ S_{{75}}&=\dfrac{{1}(1-\left({1.2}\right)^{{75}})}{1-\left({1.2}\right)} \\\\ S_{{75}}&=-5(1-\left({1.2}\right)^{{75}}) \\\\ S_{{{75}}} &\approx 4{,}340{,}731.85 \end{aligned}$ The answer $ 4{,}340{,}731.85 $